Frequently it is desirable to assign a price to an asset that is not currently traded, or to independently verify a price of an asset that is traded. There are two standard methods for assigning such prices (both of which earned Nobel prizes). The first is the capital asset pricing model (CAPM) [1] which is a method for assigning a price to an asset that has a duration of one period of time (such as one year) and is not traded at intermediate times. The second is the Black-Scholes method [2] for pricing financial derivatives, which assigns a price to a derivative (such as an option) on a underlying asset that is traded.
A derivative security is a security whose payoff is determined by the outcome of another underlying security. For example, a stock option on a traded stock is a derivative, since the final payoff of the option is completely determined by the value of the stock at the terminal time. The Black-Scholes equation is the standard method for determining the value of derivatives. It is based on the fact that, in continuous time, it is possible to replicate the payoff of a derivative by a portfolio consisting of the underlying security and a risk free asset (such as a U.S. Treasury bill) with a fixed risk free interest rate r. The fractions of the portfolio devoted to each of its components is adjusted continuously so that the portfolio's response to changes in the underlying security will perfectly mirror the response of the derivative. This adjustment process is termed a replication strategy. It is argued that the value of the derivative is equal to the cost of replication strategy. The cost V is determined by the Black-Scholes equation, which gives the cost (or value) V(x,t) for values of x≧0 and 0≦t≦T where T is the terminal time of the derivative. Specifically, the equation isrV(x, t)=Vt(x, t)+Vx(x, t)rx+½Vxx(x, t)x2σ2.   (1)Here x denotes the value of the underlying security, r is the annual risk free interest rate, and σ2 is the annual volatility of the underlying security. The notations Vt, Vx, Vxx denote, respectively, the first partial derivative with respect to t, and to x, and the second partial derivative with respect to x. The equation is solved with the boundary condition V(x, T)=F(x(T)), where F denotes the payoff at time T of the derivative. For example, if the derivative is a call option with strike price K, then F(x(T))=max (x(T)−K, 0).
A more general situation is where a payoff depends on a variable xe but this variable is not traded. For example, an option depending on a firm's revenue is of this form, because the revenue (which is xe in this case) is not traded. Hence the payoff depends on a non-traded underlying variable. In these situations it is impossible to form a replicating strategy using the conventional Black-Scholes equation because it is impossible to trade the underlying variable. The assumption underlying the conventional Black-Scholes equation breaks down.
Such problems have been studied by other researchers. The idea of using a market hedging strategy to minimize the expected squared error between the final value of the hedge and the actual payoff was proposed by Föllmer and Sondermann [3], who showed that it was possible in principle. Because minimizing the expected squared error is equivalent to orthogonal projection of the payoff onto the space of marketed payoffs (under a standard definition of projection), the method is often referred to as projection pricing. Their analysis, however, is purely abstract and does not exhibit any practical method for explicitly finding the hedge by solving a partial differential equation or a discrete version of it.
Merton (in his Nobel Prize acceptance speech [6]) emphasized the importance of the problem. He proposed a procedure based on the original Black-Scholes equation, but it is essentially an ad hoc method that does not coincide with projection. He and Pearson [7] studied a general framework of incomplete markets and formulated prices based on a consumer maximization problem. Again their method is abstract and does not provide a direct formula for the value that could be used in practice.
Some practitioners have proposed various ad hoc translations of the standard Black-Scholes equation to specific situations. For example, it is common practice to artificially increase the volatility of the underlying variable in an attempt to recognize that the variable is not really a traded asset. These methods are not based on optimality, do not fundamentally revise the original Black-Scholes equation, nor have any other real theoretical basis.
An approach that directly addresses the problem presents a solution in terms of a market price of risk, which is applicable to all derivatives of a non-marketed variable. The market price is difficult to measure, but it has sometimes been estimated [8]. However, even if the market price of risk is known, it does not lead to a hedging strategy.
Overall, there has not been an effective and practical method proposed that prices, optimally hedges, and computes the residual risk (after hedging) of derivatives of non-marketed variables.